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THE SETS of MONOMORPHISMS and of ALMOST OPEN OPERATORS BETWEEN LOCALLY CONVEX SPACES 1. Introduction and Preliminaries Bounded B

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THE SETS of MONOMORPHISMS and of ALMOST OPEN OPERATORS BETWEEN LOCALLY CONVEX SPACES 1. Introduction and Preliminaries Bounded B PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 12, Pages 3683{3690 S 0002-9939(01)06248-7 Article electronically published on June 27, 2001 THE SETS OF MONOMORPHISMS AND OF ALMOST OPEN OPERATORS BETWEEN LOCALLY CONVEX SPACES JOSE´ BONET AND JOSE´ A. CONEJERO (Communicated by Jonathan M. Borwein) Abstract. If the set of monomorphisms between locally convex spaces is not empty, then it is an open subset of the space of all continuous and linear oper- ators endowed with the topology of the uniform convergence on the bounded sets if and only if the domain space is normable. The corresponding charac- terization for the set of almost open operators is also obtained; it is related to the lifting of bounded sets and to the quasinormability of the domain space. Other properties and examples are analyzed. 1. Introduction and preliminaries Bounded below and almost open continuous linear operators between normed spaces, their relation with the topological divisors of zero in the normed algebra of all operators, and the approximate point spectrum have been extensively studied. We refer to the books of Berberian [3, Sections 56, 57], Harte [8], and the articles of Abramovich, Aliprantis and Polyrakis [2] and Harte [9]. In [2, Proposition 2.2] it is shown that the set of bounded below operators (or monomorphisms) between two normed spaces X and Y is open in the normed space of operators L(X; Y ). The corresponding result for almost open operators can be seen in [8, Theorem 3.4.3]. These results are extensions of the well-known fact (see e.g. [10, Theorem 18.12]) that the set of isomorphisms from a Banach space X onto a Banach space Y is an open subset of the Banach space L(X; Y ), a result which can be proved using the Neumann series for a linear operator. We refer to the recent article by Casazza and Kalton [5] for extensions of Paley-Wiener perturbation theory. In the case of continuous linear operators from a locally convex space E into itself, it was proved by Kasahara in 1972 in [12] that if the set of isomorphisms is open in the space of operators Lb(E) endowed with the topology of the uniform convergence on the bounded subsets of E, then the space E must be normable (see the quotation in M. Akkar [1]). If E is complete, in the terminology of topological algebras, Lb(E) is a Q-algebra if and only if E is normable. Received by the editors May 2, 2000. 2000 Mathematics Subject Classification. Primary 46A32, 46A03, 46H35, 47A05, 47L05. Key words and phrases. Bounded below operators, monomorphisms, almost open operators, locally convex spaces, quasinormable spaces. The authors were partially supported by the project DGESIC, PB97-0333. The second author was also supported by the Universidad Polit´ecnica de Valencia, grant 19980998. c 2001 American Mathematical Society 3683 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3684 JOSE´ BONET AND JOSE´ A. CONEJERO The purpose of this note is to characterize completely when the set of monomor- phisms and the set of almost open operators are open in the set of all operators L(E;F) between two locally convex spaces E and F for the topology of uniform convergence on the bounded subsets of E. The characterizations are obtained in Theorems 2.1 and 2.2. The role of the quasinormability of the space E is analyzed in Proposition 2.4 and Corollary 2.5. Theorem 2.6 studies the set of surjective and open operators between non-complete normed spaces. Several examples and propositions discuss other properties of these classes of operators. Our notation for Banach spaces, operator theory and locally convex spaces is standard and we refer to the books [3, 8, 10, 11, 15, 17, 18]. Unless explicitly mentioned, E and F denote Hausdorff locally convex spaces. If E is a locally convex space, the set of continuous seminorms on E is denoted by cs(E). If p 2 cs(E), the unit ball of this seminorm is denoted by Up := fx 2 E j p(x) ≤ 1g. The family of all closed absolutely convex neighbourhoods of the origin in E is denoted by U0(E). The Minkowski functional of U 2U0(E) is denoted by pU , and pU 2 cs(E). If E is a normed space with the norm jj:jj, its closed unit ball is denoted by UE := fx 2 E jjjxjj ≤ 1g. The set of all closed absolutely convex bounded subsets of a locally convex space E is denoted by B(E). If B 2B(E), we denote by EB the normed space given by the linear span of B endowed with the norm defined by the Minkowski functional pB of B.ThesetB 2B(E)isa Banach disc if EB is a Banach space. The space E is called locally complete if every B 2B(E) is a Banach disc. Every complete space is locally complete. As usual, the symbol Lb(E;F) denotes the locally convex space of all (always linear and continuous) operators from a locally convex space E into a locally convex space F , endowed with the locally convex topology of uniform convergence on the bounded subsets of E. A basis of neighbourhoods of zero of Lb(E;F)isgivenby the sets W(B;V ):=fT 2 L(E;F) j T (B) ⊆ V g as B runs in B(E)andV runs in U0(F ). This topology is defined by the seminorms qB(T ):=supfq(T (b)) j b 2 Bg,forB 2B(E)andq 2 cs(F ). If E and F are normed, then Lb(E;F) is the usual normed space of operators. If E = F ,wewrite Lb(E). An operator T 2 L(E;F) is called bounded below if for every p 2 cs(E)there is q 2 cs(F )withp(x) ≤ q(T (x)) for each x 2 E. Compare with the definition in the case of normed spaces in [2, Definition 2.1], [3, p. 173] and [8, Definition 3.3.1]. Clearly every bounded below operator is injective. It is easy to see that an operator T 2 L(E;F) is bounded below if and only if T is an isomorphism from E into F , i.e. T is a monomorphism in the sense of [11, vol. II, p. 2 ff.], i.e. T is linear continuous injective and open into its image. The set of all monomorphisms from E to F is denoted by mo(E;F). An operator T 2 L(E;F) is called almost open,andwewriteT 2 ao(E;F)if for every U 2U0(E)thereisV 2U0(F ) such that V ⊆ T (U), the closure taken in F . Clearly every almost open operator has dense range. Almost open operators are precisely the nearly open operators with dense range in the sense of Pt´ak (see [11, vol. II, p. 24]). The case of almost open operators between normed spaces is studied in [8, p. 65 ff.]. The class of almost open operators between Fr´echet spaces coincides with the class of surjective operators by the Banach-Schauder open mapping theorem (see [11, vol. I, p. 166] or [15, Chapter 8]). The relation between License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use OPERATORS BETWEEN LOCALLY CONVEX SPACES 3685 the properties of T 2 L(E;F) mentioned above and its transpose T t : F 0 ! E0 is analyzed in [3, 57.16{18], [8, 5.5.2{4], and [9, Section 2] for normed spaces E and F . The situation for locally convex spaces is more complicated; we refer to Dierolf and Zarnadze [6]. 2. Results Theorem 2.1. Assume that mo(E;F) is not empty. The set mo(E;F) is open in Lb(E;F) if and only if E is normable. Proof. First we suppose that mo(E;F) is open and non-empty. We select T0 2 mo(E;F), and a pair B 2B(E)andV 2U0(F ) such that every S 2 L(E;F)with (S − T0)(B) ⊆ V satisfies that S 2 mo(E;F). We claim that V \ T0(E) ⊆ T0(B), which implies that V \T0(E) is a bounded neighbourhood of zero in T0(E), whence E is normable by Kolmogorov's theorem. We prove the claim by contradiction: Suppose that there is x 2 E with T0(x) 2 V n T0(B), in particular x =6 0. We apply 0 the Hahn-Banach theorem to find v 2 F such that v(T0(x)) = 1, jv(T0(b))j < 1for all b 2 B. We define S : E ! F by S(z):=T0(z) − v(T0(z))T0(x) for every z 2 E. Clearly S 2 L(E;F). For every b 2 B we have (T0 −S)(b)=v(T0(b))T0(x) 2 V .By assumption S 2 mo(E;F). But S(x) = 0, and this implies that S is not injective, a contradiction. Now we assume that E is a normed space with a norm jj:jj, and we fix T0 2 mo(E;F). By definition, there is q 2 cs(F ) such that jjxjj ≤ q(T0(x)) for each 2 2 f − jjj jj ≤ }≤ 1 x E.IfS L(E;F) satisfies sup q(T0(x) S(x)) x 1 2 ,wehave 1 jjxjj ≤ q(T (x)) ≤ q(S(x)) + q(T (x) − S(x)) ≤ q(S(x)) + jjxjj for x 2 E: 0 0 2 This implies jjxjj ≤ 2q(S(x)) for each x 2 E and we conclude that S 2 mo(E;F).
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